Two dimensional elastic collision - unequal masses

In summary, the elastic collision of two particles with rest masses will result in the velocity of the center of mass being unchanged, but the kinetic energy and momentum of the particles will be conserved.
  • #1
Or Ozery
6
0
Hi,

I need to find an expression for u1 and u2 using m1, m2, v1 and alpha.
See attached image for more details.

Thanks in advance,
Or Ozery
 

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  • #2
I can't open the image (yet), but I'm sure you have to find expressions for the initial and final energy and momentum and use energy, momentum conservation to solve for the two unknowns.
 
  • #3
Yep

I know, I get three equations (1 from energy, and 2 from the momentum vector) with 3 variables, but they are very cumbersome - I can't find a solution...
 
  • #4
Should be enough - in the mean time, want to describe the problem so we can look at it?
 
  • #5
Consider the elastic collision of two particles with rest masses m1 and m2.
Particle 1 is moving with speed v1 and particle 2 is at rest.
We choose the coordinate system such that particle 1 is initially moving along the x axis.
The two vectors, initial and final velocity of particle 1, will define the x-y plane.
Because of conservation of momentum, the final velocity of particle 2 is also confined to the x-y plane.
After the collision particle 1 makes an angle alpha with the x-axis and its velocity is u1(cos alpha; sin alpha).

Express u1 using v1, m1, m2 and alpha.
 
  • #6
Right. So you do realize that Kinetic energy and momentum are both conserved. The velocity of the center of mass is also unchanegd.

Solving for the final velocities from using these principles... we get..

[tex] u1 = \frac{v_1(m_1 - m_2) + 2m_2v_2}{m_1 + m_2} [/tex]

[tex] u2 = \frac{v_2(m_2 - m_1) + 2m_1v_1}{m_1 + m_2} [/tex]

Hope this is of some use.
 
  • #7
But where is alpha in your formulas?
u1 and u2 depand on alpha as well...

Energy conservation:
[tex]m_{1}v_{1}^2 = m_{1}u_{1}^2 + m_{2}u_{2}^2[/tex]

Momentum conservation:
[tex]m_{1}v_{1}= m_{1}u_{1}\cos\alpha + m_{2}u_{2}\cos\beta[/tex]
[tex]m_{1}u_{1}\sin\alpha = m_{2}u_{2}\sin\beta[/tex]

Need to solve these equations (unknowns are u1, u2 and beta).
 

1. How do you calculate the final velocities of two objects after a two dimensional elastic collision with unequal masses?

To calculate the final velocities of two objects after a two dimensional elastic collision with unequal masses, you can use the equations:
V1f = (m1 - m2)/(m1 + m2) * V1i + (2 * m2)/(m1 + m2) * V2i
V2f = (2 * m1)/(m1 + m2) * V1i + (m2 - m1)/(m1 + m2) * V2i
where m1 and m2 are the masses of the two objects, V1i and V2i are the initial velocities of the objects, and V1f and V2f are the final velocities of the objects.

2. What is the difference between an elastic collision and an inelastic collision?

An elastic collision is a type of collision in which both kinetic energy and momentum are conserved. This means that the total kinetic energy before the collision is equal to the total kinetic energy after the collision, and the total momentum before the collision is equal to the total momentum after the collision. In contrast, an inelastic collision is a type of collision in which kinetic energy is not conserved. Some of the kinetic energy is lost during the collision, usually in the form of heat or sound.

3. How is the angle of deflection calculated in a two dimensional elastic collision?

The angle of deflection in a two dimensional elastic collision is calculated using the equation:
θ = tan^-1((V1i * sin α1 + V2i * sin α2)/(V1i * cos α1 - V2i * cos α2))
where V1i and V2i are the initial velocities of the two objects, and α1 and α2 are the angles at which the objects are initially moving.

4. Can the kinetic energy of an object decrease after a two dimensional elastic collision?

No, the kinetic energy of an object cannot decrease after a two dimensional elastic collision. In an elastic collision, the total kinetic energy is conserved, meaning that it remains the same before and after the collision. However, the kinetic energy of each individual object may change as a result of the collision.

5. What are some real-life examples of two dimensional elastic collisions with unequal masses?

Some real-life examples of two dimensional elastic collisions with unequal masses include billiard balls colliding on a pool table, two cars colliding at an intersection, or a tennis ball hitting a tennis racket. In all of these scenarios, the objects have different masses and collide with each other, resulting in a change in their velocities and directions.

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